### AG Differentialgeometrie (Summer Semester 2019)

- Lecturer: Prof. Dr. Wilderich Tuschmann
- Classes: Seminar (0176100)
- Weekly hours: 2

Schedule | ||
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Seminar: | Thursday 15:45-17:15 | SR 2.66 |

Lecturers | ||
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Lecturer | Prof. Dr. Wilderich Tuschmann | |

Office hours: by appointment | ||

Room 1.002 Kollegiengebäude Mathematik (20.30) | ||

Email: tuschmann@kit.edu |

**Talks**

** 25.4.2019**

**Georg Frenck (Universität Münster)**

* The action of Diff(M) on the space of psc metrics*

In 1974, Nigel Hitchin showed that the orbit map associated to the action of the diffeomorphism group of the d-dimensional sphere on the space of its positive scalar curvature metrics induces a nontrivial map on components provided that d = 0,1 mod 8. Hence, this map detects non-isotopic psc metrics.

In this talk, I will present a rigidity result for this action. Among other applications, it implies that the orbit map for the sphere is trivial in every other high enough dimension, so Hitchin’s detection result is the only possible one of this kind for high-dimensional spheres.

** 9.5.2019**

**Oliver Goertsches (Philipps-Universität Marburg)**

* Biquotients in symplectic geometry*

Biquotients are generalizations of homogeneous spaces that naturally occur in Riemannian geometry, mostly in regards to questions concerning positive and nonnegative curvature. In this talk we will explain why they are also of interest in symplectic and Kähler geometry, by constructing many such structures on equal rank biquotients. Particular emphasis will be put on Eschenburg's twisted flag manifold SU(3)//T, which we will compare to Tolman's and Woodward's examples of symplectic manifolds admitting Hamiltonian non-Kähler torus actions. (This is joint work with Panagiotis Konstantis and Leopold Zoller.)

** 16.5.2019**

**Joachim Lohkamp (Universität Münster)**

* Hyperbolic unfoldingss of minimal hypersurfaces*

Minimal Hypersurfaces may carry delicate singularities and both the regular

part of the surface but also the elliptic analysis on such surfaces degenerate towards these

singularities. Nevertheless, we get a surprisingly fine control over the asymptotic analysis of

elliptic equations on these surfaces. This is owing to canonical conformal deformations of the

regular part of the surface to complete Gromov hyperbolic spaces of bounded geometry,

the hyperbolic unfoldings. We introduce to these unfoldings and explain some striking

geometro-analytic applications in potential theory and in scalar curvature geometry.

** 27.6.2019**

**Artem Nepechiy (Universität Köln)**

* Toward canonical convex functions in Alexandrov spaces*

For a point $p$ in a Riemannian manifold all small metric balls around $p$ are convex. This is no longer true in the realm of Alexandrov spaces. In particular it might happen that the distance squared function at $p$ is not convex in any neighborhood around $p$.

In this talk I will explain how to construct for every point $p$ in a finite-dimensional Alexandrov space a function in a small neighborhood around p, which approximates the distance squared function at $p$ up to second order and has convexity properties as we would expect it from Euclidean space. Moreover, the function $f$ can be lifted to Gromov-Hausdorff close Alexandrov spaces of the same dimension.

** 4.7.2019**

**Ana Karla Garcia (KIT)**

*Moduli space of flat metrics*

We are going to introduce a result of Wolf's that tells us how to compute the moduli space of flat metrics of a flat manifold. Then we will consider the case of 3-dimensional flat manifolds, where all the moduli spaces of flat metrics are contractible except for two. We will conclude with some comments on the 4-dimensional case.

** 18.7.2019**

**Wolfgang Heil (Florida State University)**

*2-stratifolds and 3-manifolds*

2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed branch curves. Since every 3-manifold has some 2-complex as a spine one may ask which 3-manifolds have 2-stratifold spines. In this talk we will define 2-stratifolds and obtain a list of all closed 3-manifolds with 2-stratifold spines. This is joint work with J. C. Gómez-Larrañaga and F. González-Acuña.

** 25.7.2019**

**Jian Ge (Peking University)**

*Fillings of Alexandrov spaces*

In this talk, we will talk about the geometric properties of Alexandrov spaces that bound a given positively curved space. Under certain assumptions on the boundary, we get rigidity results. Part of the talk is based on joint works with Ronggang Li.

**Note: The talk will start at 4:15PM.**